Proposition 4.8.8.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:
The functor $F$ is essentially $n$-categorical.
The comparison map $F': \operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ of Remark 4.8.8.15 is an equivalence of $\infty $-categories.
Proof. It follows from Proposition 4.8.8.14 (and Proposition 4.8.6.35) that the comparison map $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ is essentially $n$-categorical. By virtue of Remark 4.8.6.8, we can replace $(1)$ by the following condition:
The functor $F'$ is essentially $n$-categorical: that is, it is $m$-full for $m \geq n+2$.
Since $F'$ is also $m$-full for $m \leq n+1$ (Corollary 4.8.8.19), the equivalence $(1') \Leftrightarrow (2)$ follows from Remark 4.8.5.11. $\square$